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Full text of "Treatise On Analysis Vol-Ii"

402       XV   NORMED ALGEBRAS AND SPECTRAL THEORY

In particular:

(15.11.12)    If H is any positive, self-adjoint operator then there exists a
unique-positive self-adjoint operator H1 such that H'2 = H.

Apply (15.11.11) with M = N = R+ and/(C) = C2.

The unique positive self-adjoint operator H' defined in (15.11.12) is
denoted by H112.

Example

(15.11.13)    Let E be the Hilbert space Lc(R; A), where A is Lebesgue measure.
Since the function /() = e~^ *s A-integrable, the convolution g*-*f*g
defines, on passing to the equivalence classes, a continuous operator H on E
(14.10.6) with norm N^/) =2. As in (11.6.1), it is immediately seen that H
is self-adjoint. It can be shown directly (Problem 5) that the interval [0, 2] in
R is equal to Sp(H); this also follows from the general theorems of harmonic
analysis (Chapter XXII).

Note that, for each a e R, the function ga(t) = eiat is such that the con-
volution/*^ is defined and equal to 2gJ(l + a2). However, it is not the
case that the ga are " eigenfunctions " of H, because they do not belong to
&c(R, A). In Chapter XXIII we shall obtain a generalization and an interpre-
tation of this phenomenon.

(15.11.14)    The case of normal operators whose spectrum contains no non-
isolated point 7^0. In this case (which is that of compact normal operators
(11.4.1)) let (An) be the (finite or infinite) sequence of points of Sp(N), other
than 0. These are the eigenvalues of N (15.11.6). The eigenspace E(AW; N)
corresponding to AM is just the space E({AW}) defined in (15.11.9), and these
closed subspaces are therefore pairwise orthogonal. Moreover, we have
E({0}) = Ker(N), the spectrum of the restriction of N to this subspace being
reduced to 0 (15.11.9). Finally, E is the Hilbert sum of E({0}) and the E({AW}).
For it is enough to apply (15.11.8(iv)) to the sequence of functions (/),
where /n(Q =1   for   = Ak  with  kn, /() = 0  for C = Ak and k > n,
and /n(0) = 1; this shows in particular that every x e E is the limit of the
sum of its projections on E({0}) and the E({Ak}) with k < n. Hence the
result (6.4).

In particular, if E is ./mfte-dimensional, a normal operator on E may be
defined to be an operator whose matrix is diagonal, with respect to a suitably
chosen orthonormal basis of E.must hav0^^ ^ which is the inverse of/>then bY